Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case
This is the third in a series of five papers studying special Lagrangian submanifolds (SLV m-folds) X in (almost) Calabi-Yau m-folds M with singularities x 1,..., x n locally modelled on special Lagrangian cones C 1,..., C n in ℂ m with isolated singularities at 0. Readers are advised to begin with...
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| Format: | Journal article |
| Language: | English |
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2004
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| _version_ | 1826286446870265856 |
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| author | Joyce, D |
| author_facet | Joyce, D |
| author_sort | Joyce, D |
| collection | OXFORD |
| description | This is the third in a series of five papers studying special Lagrangian submanifolds (SLV m-folds) X in (almost) Calabi-Yau m-folds M with singularities x 1,..., x n locally modelled on special Lagrangian cones C 1,..., C n in ℂ m with isolated singularities at 0. Readers are advised to begin with Paper V. This paper and Paper IV construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds Ñ t in M for small t > 0. Suppose L 1,..., L n are Asymptotically Conical SL m-folds in ℂ m, with L i asymptotic to the cone C i at infinity. We shrink L i by a small t > 0, and glue t L i into X at x i for i = 1,..., n to get a 1-parameter family of compact, nonsingular Lagrangian m-folds N t for small t > 0. Then we show using analysis that when t is sufficiently small we can deform N t to a compact, nonsingular special Lagrangian m-fold Ñ t, via a small Hamiltonian deformation. This \tilde Ñ t depends smoothly on t, and as t → 0 it converges to the singular SL m-fold X, in the sense of currents. This paper studies the simpler cases, where by topological conditions on X and L i we avoid various obstructions to existence of Ñ t. Paper IV will consider more complex cases when these obstructions are nontrivial, and also desingularization in families of almost Calabi-Yau m-folds. |
| first_indexed | 2024-03-07T01:43:54Z |
| format | Journal article |
| id | oxford-uuid:97c372b6-55c7-4a49-a4e3-58ea47bd0de7 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:43:54Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | oxford-uuid:97c372b6-55c7-4a49-a4e3-58ea47bd0de72022-03-27T00:02:11ZSpecial Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed caseJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:97c372b6-55c7-4a49-a4e3-58ea47bd0de7EnglishSymplectic Elements at Oxford2004Joyce, DThis is the third in a series of five papers studying special Lagrangian submanifolds (SLV m-folds) X in (almost) Calabi-Yau m-folds M with singularities x 1,..., x n locally modelled on special Lagrangian cones C 1,..., C n in ℂ m with isolated singularities at 0. Readers are advised to begin with Paper V. This paper and Paper IV construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds Ñ t in M for small t > 0. Suppose L 1,..., L n are Asymptotically Conical SL m-folds in ℂ m, with L i asymptotic to the cone C i at infinity. We shrink L i by a small t > 0, and glue t L i into X at x i for i = 1,..., n to get a 1-parameter family of compact, nonsingular Lagrangian m-folds N t for small t > 0. Then we show using analysis that when t is sufficiently small we can deform N t to a compact, nonsingular special Lagrangian m-fold Ñ t, via a small Hamiltonian deformation. This \tilde Ñ t depends smoothly on t, and as t → 0 it converges to the singular SL m-fold X, in the sense of currents. This paper studies the simpler cases, where by topological conditions on X and L i we avoid various obstructions to existence of Ñ t. Paper IV will consider more complex cases when these obstructions are nontrivial, and also desingularization in families of almost Calabi-Yau m-folds. |
| spellingShingle | Joyce, D Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case |
| title | Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case |
| title_full | Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case |
| title_fullStr | Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case |
| title_full_unstemmed | Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case |
| title_short | Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case |
| title_sort | special lagrangian submanifolds with isolated conical singularities iii desingularization the unobstructed case |
| work_keys_str_mv | AT joyced speciallagrangiansubmanifoldswithisolatedconicalsingularitiesiiidesingularizationtheunobstructedcase |